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FinancialDerivatives
Thisbookoffersasuccinctaccountoftheprinciplesoffinancialderiva-
tivespricing.Thefirstchapterprovidesreaderswithanintuitiveexpo-
sition of basic random calculus. Concepts such as volatility and time,
random walks, geometric Brownian motion, and Itoˆ’s lemma are dis-
cussedheuristically.Thesecondchapterdevelopsgenericpricingtech-
niquesforassetsandderivatives,determiningthenotionofastochastic
discount factor or pricing kernel, and then uses this concept to price
conventionalandexoticderivatives.Thethirdchapterappliesthepric-
ingconceptstothespecialcaseofinterestratemarkets,namely,bonds
and swaps, and discusses factor models and term-structure-consistent
models.Thefourthchapterdealswithavarietyofmathematicaltopics
thatunderliederivativespricingandportfolioallocationdecisions,such
asmean-revertingprocessesandjumpprocesses,anddiscussesrelated
toolsofstochasticcalculus,suchasKolmogorovequations,martingales
techniques,stochasticcontrol,andpartialdifferentialequations.
Jamil Baz is the chief investment strategist of GLG, a London-based
hedgefund.Priortoholdingthisposition,hewasaportfoliomanager
withPIMCOinLondon,amanagingdirectorintheProprietaryTrad-
ingGroupofGoldmanSachs,chiefinvestmentstrategistofDeutsche
Bank,andexecutivedirectorofLehmanBrothersfixedincomeresearch
division.Dr.BazteachesfinancialeconomicsatOxfordUniversity.He
hasdegreesfromtheLondonSchoolofEconomics(M.Sc.),MIT(S.M.),
andHarvardUniversity(A.M.,Ph.D.).
GeorgeChackoischiefinvestmentofficerofAuda,aglobalassetman-
agementfirm,inNewYork.HeisalsoaprofessoratSantaClaraUni-
versity, California, where he teaches finance. Dr. Chacko previously
servedfortenyearsasaprofessoratHarvardBusinessSchoolinthe
financedepartment.Dr.Chackoheldmanagingdirectorshipsinfixed
incomesalesandtradingatStateStreetBankinBostonandinpension
assetmanagementatIFLinNewYork.HeholdsaB.S.fromMIT,an
M.B.A.fromtheUniversityofChicago,andanM.A.andPh.D.from
HarvardUniversity.
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Financial Derivatives
Pricing, Applications, and Mathematics
JAMIL BAZ
GLG
GEORGE CHACKO
Auda
iii
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi
Cambridge University Press
32 Avenue of the Americas, New York, NY10013-2473, USA
www.cambridge.org
Information on this title: www.cambridge.org/9780521815109
©Jamil Baz and George Chacko 2004
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2004
First paperback edition 2009
Printed in the United States of America
Acatalog recordfor this publication is available from the British Library.
Library of Congress Cataloging in Publication Data
Baz, Jamil
Financial derivatives : pricing, applications, and mathematics / Jamil Baz,
George Chacko.
p. cm.
Includes bibliographical references and index.
ISBN 0-521-81510-X
1. Derivative securities. I. Chacko, George. II. Title.
HG6024.A3B396 2003
332.63'2 – dc21
2002041452
ISBN 978-0-521-81510-9 hardback
ISBN 978-0-521-06679-2 paperback
Cambridge University Press has no responsibility for
the persistence or accuracy of URLs for external or
third-party Internet Web sites referred to in this publication
and does not guarantee that any content on such
Web sites is, or will remain, accurate or appropriate.
CY260/Baz-FM CY260/Baz 052181510X September16,2003 12:50 CharCount=0
ToMauriceandElena J.B.
Tomyparents G.C.
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Contents
Acknowledgments pagexi
Introduction 1
1 PreliminaryMathematics 5
1.1 RandomWalk 5
1.2 AnotherTakeonVolatilityandTime 8
1.3 AFirstGlanceatItoˆ’sLemma 9
1.4 ContinuousTime:BrownianMotion;More
onItoˆ’sLemma 11
1.5 Two-DimensionalBrownianMotion 14
1.6 BivariateItoˆ’sLemma 15
1.7 ThreeParadoxesofFinance 16
1.7.1 Paradox1:Siegel’sParadox 16
1.7.2 Paradox2:TheStock,Free-LunchParadox 18
1.7.3 Paradox3:TheSkillVersusLuckParadox 19
2 PrinciplesofFinancialValuation 22
2.1 Uncertainty,UtilityTheory,andRisk 22
2.2 RiskandtheEquilibriumPricingofSecurities 28
2.3 TheBinomialOption-PricingModel 41
2.4 LimitingOption-PricingFormula 46
2.5 Continuous-TimeModels 47
2.5.1 TheBlack-Scholes/MertonModel–Pricing
KernelApproach 48
2.5.2 TheBlack-Scholes/MertonModel–
ProbabilisticApproach 57
2.5.3 TheBlack-Scholes/MertonModel–Hedging
Approach 61
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viii Contents
2.6 ExoticOptions 63
2.6.1 DigitalOptions 64
2.6.2 PowerOptions 65
2.6.3 AsianOptions 67
2.6.4 BarrierOptions 71
3 InterestRateModels 78
3.1 InterestRateDerivatives:NotSoSimple 78
3.2 BondsandYields 80
3.2.1 PricesandYieldstoMaturity 80
3.2.2 DiscountFactors,Zero-CouponRates,and
CouponBias 82
3.2.3 ForwardRates 85
3.3 NaiveModelsofInterestRateRisk 88
3.3.1 Duration 88
3.3.2 Convexity 99
3.3.3 TheFreeLunchintheDurationModel 104
3.4 AnOverviewofInterestRateDerivatives 108
3.4.1 BondswithEmbeddedOptions 109
3.4.2 ForwardRateAgreements 110
3.4.3 EurostripFutures 112
3.4.4 TheConvexityAdjustment 113
3.4.5 Swaps 118
3.4.6 CapsandFloors 120
3.4.7 Swaptions 121
3.5 YieldCurveSwaps 122
3.5.1 TheCMSSwap 122
3.5.2 TheQuantoSwap 127
3.6 FactorModels 131
3.6.1 AGeneralSingle-FactorModel 131
3.6.2 TheMertonModel 135
3.6.3 TheVasicekModel 139
3.6.4 TheCox-Ingersoll-RossModel 142
3.6.5 Risk-NeutralValuation 144
3.7 Term-Structure-ConsistentModels 147
3.7.1 “Equilibrium”Versus“Fitting” 147
3.7.2 TheHo-LeeModel 153
3.7.3 TheHo-LeeModelwithTime-Varying
Volatility 157
3.7.4 TheBlack-Derman-ToyModel 162
3.8 RiskyBondsandTheirDerivatives 166
3.8.1 TheMertonModel 167
3.8.2 TheJarrow-TurnbullModel 168
